How display and use operator properties in a Python Blender UI panel?
You usually run an operator and adjust the settings afterwards in the Redo panel, or the operator should prompt the user before execution using invoke_props_dialog():

import bpy

class SimpleOperator(bpy.types.Operator):
    bl_label = "Simple Operator"
    bl_idname = "object.simple_operator"

    my_enum = bpy.props.EnumProperty(
        items=(
            ('ONE', "One", ""),
            ('TWO', "Two", ""),
            ('THREE', "Three", "")
        ),
        default='ONE'
    )

    def invoke(self, context, event):
        wm = context.window_manager
        return wm.invoke_props_dialog(self)

    def execute(self, context):
        self.report({'INFO'}, self.my_enum)
        return {'FINISHED'}


def register():
    bpy.utils.register_module(__name__)

def unregister():
    bpy.utils.unregister_module(__name__)

if __name__ == "__main__":
    register()

    bpy.ops.object.simple_operator('INVOKE_DEFAULT')


If you really want to display UI elements in a panel to be set before clicking the button, you will have to register global properties and pass them to the according operator properties (you can't add operator properties to a panel).

import bpy

class SimpleOperator(bpy.types.Operator):
    bl_label = "Simple Operator"
    bl_idname = "object.simple_operator"

    my_enum = bpy.props.EnumProperty(
        items=(
            ('ONE', "One", ""),
            ('TWO', "Two", ""),
            ('THREE', "Three", "")
        ),
        default='ONE'
    )

    def execute(self, context):
        self.report({'INFO'}, self.my_enum)
        return {'FINISHED'}


class HelloWorldPanel(bpy.types.Panel):
    """Creates a Panel in the Object properties window"""
    bl_label = "Hello World Panel"
    bl_idname = "OBJECT_PT_hello"
    bl_space_type = 'PROPERTIES'
    bl_region_type = 'WINDOW'
    bl_context = "object"

    def draw(self, context):
        layout = self.layout

        ob = context.object

        row = layout.row()
        row.prop(ob, "my_global_enum", expand=True)

        props = layout.operator(SimpleOperator.bl_idname)
        props.my_enum = ob.my_global_enum


def register():
    bpy.utils.register_module(__name__)

    bpy.types.Object.my_global_enum = bpy.props.EnumProperty(
        items=(
            ('ONE', "One", ""),
            ('TWO', "Two", ""),
            ('THREE', "Three", "")
        ),
        default='TWO'
    )


def unregister():
    bpy.utils.unregister_module(__name__)
    del bpy.types.Object.my_global_enum


if __name__ == "__main__":
    register()